We characterize when two conic curved creases are compatible with each other,
when the rule lines must converge to conic foci and reflect at the crease.
Namely, two conics are compatible (can be connected by rule segments in a
foldable curved crease pattern) if and only if they have equal or reciprocal
eccentricity. Thus, circles (eccentricity 0) and parabolas
(eccentricity 1) are compatible with only themselves (when scaled
from a focus), and ellipses (eccentricity strictly between 0 and 1)
and hyperbolas (eccentricity above 1) are compatible with themselves
and each other (but only in specific pairings).
The foundation of this result is a general condition relating
any two curved creases connected by rule segments.
We also use our characterization to analyze several curved crease designs.